CHAP, in] FERMAT'S AND WILSON'S THEOREMS. 75 C. A. Laisant and E. Beaujeux91 used the period ax . . .an of the periodic fraction to base B for the irreducible fraction Pi/q, where q is prime to B. If p2, • • - , Pn are the successive remainders, Starting with the second equation, we obtain the period a2. . .anax for p2/q. Similarly for pz/q,. . ., pn/q. Thus the /=<?(#) irreducible, fractions with denominator q separate into sets of n each. Hence f=kn. Since Bn=l, Bf=l (mod 5). L. Ottinger92 employed differential calculus to show that, in P=(a+d)(a+2d). . . \a+(p-l)d\ =a*-l+Cl»-lap -tic,-*.* ( Cr being the sum of the products of 1, 2, . . . , k taken r at a time. Hence, if p is a prime, C?"1 (r = 1, . . . , p — 2) is divisible by p, and P=a*-l+d'2d. . .(p-l)d (mod p). For a = d = l, this gives 0=l + (p — 1)! (mod p). H. Anton93 gave Gauss'28 proof of Wilson's theorem. J. Petersen94 proved Wilson's theorem, by dividing the circumference of a circle into p equal parts, where p is a prime, and marking the points 1, . . . , p. Designate by 12 . . .p the polygon obtained by joining 1 with 2, 2 with 3, . . ., p with 1. Rearranging these numbers we obtain new polygons, not all convex. While there are pi rearrangements, each polygon can be designated in 2p ways [beginning with any one of the p numbers as first and reading forward or backward], so that we get (p — 1)1/2 figures. Of these |(p — 1) are regular. The others are congruent in sets of p, since by rotation any one of them assumes p positions. Hence p divides (p — 1)1/2 -(p-l)/2 and hence (p-2)!-l. Cf. Cayley101. To prove Fermat's theorem, take p elements from q with repetitions in all ways, that is, in qp ways. The q sets with elements all alike are not changed by a cyclic pennutation of the elements, while the remaining qp— q sets are permuted in sets of p. Hence p divides (f — q. [Cf. Perott,126 Bricard.131] F. Unferdinger95 proved by use of series of exponentials that "Nouv. Ann. Math., (2), 7, 1868, 292-3. "Archiv Math. Phys., 48, 1868, 159-185. 83/6wi., 49, 1869, 297-8. MTidsskrift for Mathematik, (3), 2, 1872, 64-65 (Danish). MS;t*iincrHhprirht,fl Ak. Wisfl Winn. ft7. 1873 TT. 3ft3.